Astronomy midterm question: example of a cool star more luminous than a hot star?

Astronomy 210 Midterm 2, fall semester 2013
Cuesta College, San Luis Obispo, CA

An astronomy question on an online discussion board[*] was asked and answered:

y2j: What is an example of how a cool star can be more luminous than a hot star?
Brnt: Two stars in the constellation Orion: Betelgeuse, a supergiant, and Bellatrix, a giant.

Discuss why this answer is correct, and how you know this. Explain using Wien's law, the Stefan-Boltzmann law and/or an H-R diagram.

[*] answers.yahoo.com/question/index?qid=20080210061810AAHbs0w.

Solution and grading rubric:

• p:
  Correct. Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to show specifically how Betelgeuse (supergiant) is cooler, larger and more luminous than Bellatrix (giant), which is hotter, smaller, and less luminous.
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. As (p), but arbitrarily picks one other star to compare (typically a white dwarf) with either Betelgeuse or Bellatrix. Or instead may conflate mass with size.
• t: Contains right ideas, but discussion is unclear/incomplete or contains major errors.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least attempts to use Wien's law, H-R diagram and/or the Stefan-Boltzman law.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Discussion not based on Wien's law, H-R diagram and/or the Stefan-Boltzman law.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 70160
Exam code: midterm02nI5o
p: 13 students
r: 5 student
t: 3 students
v: 5 students
x: 1 student
y: 1 student
z: 0 students

A sample "p" response (from student 3055):
A sample "y" response (from student 1112):